Curve modifier doesn't do anything
It appears that every vertex of the curve has a double that isn't connected. Go into edit mode on the curve, select one of the connected vertices(not the handle) and press Ctrl+L to select the linked ones, then Ctrl+I to invert the selection to get the disconnected vertices, and then delete them. 

For some reason the array modifier still doesn't work with the fit curve option, but the simple alternative is to add an array modifier with fixed count and then a curve modifier below that. 

You may also want to make the rna mesh object contain only a small number of nucleotides so that you can make the rna fit the length of the curve better.